calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaextrema, the Euler–Lagrange equation is useful for solving

optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...

problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

, stating that at any point where a differentiable function attains a local extremum its

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

is zero. In

Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...

, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange equations. In

classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

, it is equivalent to

Newton's laws of motion Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...

; indeed, the Euler-Lagrange equations will produce the same equations as Newton's Laws. This is particularly useful when analyzing systems whose force vectors are particularly complicated. It has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In

classical field theory A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called qua ...

there is an analogous equation to calculate the dynamics of a field.

History

The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the

tautochrone A tautochrone curve or isochrone curve () is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point on the curve. The curve is a cycloid, and the time ...

problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to

mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...

, which led to the formulation of

Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...

. Their correspondence ultimately led to the

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

, a term coined by Euler himself in 1766.

Statement

Let

(X,L)

be a real dynamical system with

n

degrees of freedom. Here

X

is the configuration space and

L=L(t,(t), (t))

the '' Lagrangian'', i.e. a smooth

real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real ...

such that

(t) \in X,

and

(t)

is an

n

-dimensional "vector of speed". (For those familiar with

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

,

X

is a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...

, and

L : _t \times X \times TX \to ,

where

TX

is the

tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...

of

X).

Let

(a,b,\boldsymbol x_a,\boldsymbol x_b)

be the set of smooth paths

\to X

for which

\boldsymbol q(a) = \boldsymbol x_a

and

\boldsymbol q(b) = \boldsymbol x_b.

The action functional

S : (a,b,\boldsymbol x_a,\boldsymbol x_b) \to \mathbb

is defined via

= \int_a^b L(t,\boldsymbol q(t),\dot(t))\, dt.

A path

\boldsymbol q \in (a,b,\boldsymbol x_a,\boldsymbol x_b)

is a

stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...

of

S

if and only if Here,

\dot(t)

is the

time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...

of

\boldsymbol q(t).

When we say stationary point, we mean a stationary point of

S

with respect to any small perturbation in

\boldsymbol q

. See proofs below for more rigorous detail.

Example

A standard example is finding the real-valued function ''y''(''x'') on the interval 'a'', ''b'' such that ''y''(''a'') = ''c'' and ''y''(''b'') = ''d'', for which the

path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desir ...

length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...

along the

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

traced by ''y'' is as short as possible. :

\text = \int_^ \sqrt = \int_^ \sqrt\,\mathrmx,

the integrand function being

L(x,y, y') = \sqrt

. The partial derivatives of ''L'' are: :

\frac = \frac \quad \text \quad
\frac = 0.

By substituting these into the Euler–Lagrange equation, we obtain :

\begin
\frac \frac &= 0 \\ 
\frac &= C = \text \\
\Rightarrow y'(x)&= \frac =: A \\
\Rightarrow y(x) &= Ax + B
\end

that is, the function must have a constant first derivative, and thus its

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

is a straight line.

Generalizations

Single function of single variable with higher derivatives

The stationary values of the functional :

= \int_^ \mathcal(x, f, f', f'', \dots, f^)~\mathrmx ~;~~ f' := \cfrac, ~f'' := \cfrac, ~ f^ := \cfrac

can be obtained from the Euler–Lagrange equation :

\cfrac - \cfrac\left(\cfrac\right) + \cfrac\left(\cfrac\right) - \dots +
  (-1)^k \cfrac\left(\cfrac\right)  = 0

under fixed boundary conditions for the function itself as well as for the first

k-1

derivatives (i.e. for all

f^, i \in \

). The endpoint values of the highest derivative

f^

remain flexible.

Several functions of single variable with single derivative

If the problem involves finding several functions (

f_1, f_2, \dots, f_m

) of a single independent variable (

x

) that define an extremum of the functional :

= \int_^ \mathcal(x, f_1, f_2, \dots, f_m, f_1', f_2', \dots, f_m')~\mathrmx ~;~~ f_i' := \cfrac

then the corresponding Euler–Lagrange equations are :

\begin
     \frac - \frac\left(\frac\right) = 0 ; \quad i = 1, 2, ..., m
   \end

Single function of several variables with single derivative

A multi-dimensional generalization comes from considering a function on n variables. If

\Omega

is some surface, then :

= \int_ \mathcal(x_1, \dots , x_n, f, f_, \dots , f_)\, \mathrm\mathbf\,\! ~;~~ f_ := \cfrac

is extremized only if ''f'' satisfies the

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

:

\frac - \sum_^ \frac\left(\frac\right) = 0.

When ''n'' = 2 and functional

\mathcal I

is the energy functional, this leads to the soap-film

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

problem.

Several functions of several variables with single derivative

If there are several unknown functions to be determined and several variables such that :

= \int_ \mathcal(x_1, \dots , x_n, f_1, \dots, f_m, f_, \dots , f_, \dots, f_, \dots, f_) \, \mathrm\mathbf\,\! ~;~~ f_ := \cfrac

the system of Euler–Lagrange equations is :

\begin
    \frac - \sum_^ \frac\left(\frac\right) &= 0_1 \\
    \frac - \sum_^ \frac\left(\frac\right) &= 0_2 \\
    \vdots \qquad \vdots \qquad &\quad \vdots  \\
    \frac - \sum_^ \frac\left(\frac\right) &= 0_m.
  \end

Single function of two variables with higher derivatives

If there is a single unknown function ''f'' to be determined that is dependent on two variables ''x''₁ and ''x''₂ and if the functional depends on higher derivatives of ''f'' up to ''n''-th order such that :

& = \int_ \mathcal(x_1, x_2, f, f_, f_, f_, f_, f_, \dots, f_)\, \mathrm\mathbf \\ & \qquad \quad f_ := \cfrac \; , \quad f_ := \cfrac \; , \;\; \dots \end

then the Euler–Lagrange equation is :

\begin
    \frac
    & - \frac\left(\frac\right)
      - \frac\left(\frac\right) 
      + \frac\left(\frac\right)
      + \frac\left(\frac\right)
      + \frac\left(\frac\right) \\
    & - \dots
      + (-1)^n \frac\left(\frac\right) = 0
  \end

which can be represented shortly as: :

\frac +\sum_^n \sum_ (-1)^j \frac \left( \frac\right)=0

wherein

\mu_1 \dots \mu_j

are indices that span the number of variables, that is, here they go from 1 to 2. Here summation over the

\mu_1 \dots \mu_j

indices is only over

\mu_1 \leq \mu_2 \leq \ldots \leq \mu_j

in order to avoid counting the same

partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...

multiple times, for example

f_ = f_

appears only once in the previous equation.

Several functions of several variables with higher derivatives

If there are ''p'' unknown functions ''f''_i to be determined that are dependent on ''m'' variables ''x''₁ ... ''x''_m and if the functional depends on higher derivatives of the ''f''_i up to ''n''-th order such that :

& = \int_ \mathcal(x_1, \ldots, x_m; f_1,\ldots,f_p; f_,\ldots, f_; f_,\ldots, f_;\ldots; f_, \ldots, f_)\, \mathrm\mathbf \\ & \qquad \quad f_ := \cfrac \; , \quad f_ := \cfrac \; , \;\; \dots \end

where

\mu_1 \dots \mu_j

are indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is :

\frac +\sum_^n \sum_ (-1)^j \frac \left( \frac\right)=0

where the summation over the

\mu_1 \dots \mu_j

is avoiding counting the same derivative

f_ = f_

several times, just as in the previous subsection. This can be expressed more compactly as :

\sum_^n \sum_ (-1)^j \partial_^j \left( \frac\right)=0

Field theories

Generalization to manifolds

Let

M

be a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...

, and let

C^\infty(,b

denote the space of smooth functions

f\colon,b to M

. Then, for functionals

\to \mathbb

of the form :

\int_a^b (L\circ\dot)(t)\,\mathrm t

where

L\colon TM\to\mathbb

is the Lagrangian, the statement

\mathrm S_f=0

is equivalent to the statement that, for all

t\in,b /math>, each coordinate frame trivialization (x^i,X^i) of a neighborhood of \dot(t) yields the following \dim M equations:
: \forall i:\frac\frac\bigg, _=\frac\bigg, _. Euler-Lagrange equations can also be written in a coordinate-free form as : \mathcal_\Delta \theta_L=dL where \theta_L is the canonical momenta

1-form In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the t ...

corresponding to the Lagrangian

L

. The vector field generating time translations is denoted by

\Delta

and the

Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...

is denoted by

\mathcal

. One can use local charts

(q^\alpha,\dot^\alpha)

in which

\theta_L=\fracdq^\alpha

and

\Delta:=\frac=\dot^\alpha\frac+\ddot^\alpha\frac

and use coordinate expressions for the Lie derivative to see equivalence with coordinate expressions of the Euler Lagrange equation. The coordinate free form is particularly suitable for geometrical interpretation of the Euler Lagrange equations.

Notes

References

* * * * * Roubicek, T.:
Calculus of variations
Chap.17 in
Mathematical Tools for Physicists
(Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, , pp. 551–588. {{DEFAULTSORT:Euler-Lagrange Equation Eponymous equations of mathematics Eponymous equations of physics Ordinary differential equations Partial differential equations Calculus of variations Articles containing proofs Leonhard Euler

History

Statement

Example

Generalizations

Single function of single variable with higher derivatives

Several functions of single variable with single derivative

Single function of several variables with single derivative

Several functions of several variables with single derivative

Single function of two variables with higher derivatives

Several functions of several variables with higher derivatives

Field theories

Generalization to manifolds

See also

Notes

References